I need to show the following inequality:
$$\sum_{j=1}^n \left(\frac{D_j}{Q_j} \left\lceil\frac{Q_j}{T_c}\right\rceil \right) \ge \frac{\sum_{j=1}^n D_j}{\sum_{j=1}^n Q_j} \left\lceil\frac{\sum_{j=1}^n Q_j}{T_c}\right\rceil$$
in which, we know the followings: $D_j \ge Q_j$,
$D_j$ are all integer, and same as $T_c$, which is a constant term.
The inequality need not hold. Take $n=2,D_1=3,Q_1=2,D_2=4,Q_2=3,T_c=2.$ Then using $c(x)$ for the ceiling of $x$, the inequality in this case would say $$(3/2)c(2/2)+(4/3)c(3/2) \ge (3+4)/(2+3)c((2+3)/2).$$ But the left side is $3/2+(4/3)\cdot 2=25/6=4.1666..,$ while the right side is $(7/5)\cdot 3=21/5=4.20.$
I tested and found lots of cases in the $n=2$ case where the inequality fails. The above was just a small one for which we have each $D_i$ greater than the corresponding $Q_i$.